A381337 -id:A381337 - cp's OEIS frontend

A382297 Indices of right triangles in A381337.

1, 2, 3, 4, 6, 7, 12, 14, 17, 23, 28, 31, 34, 35, 49, 51, 62, 69, 71, 73, 77, 85, 93, 97, 98, 102, 119, 127, 142, 161, 170, 194, 196, 199, 223, 233, 238, 241, 245, 279, 281, 287, 291, 337, 357, 381, 388, 391, 398, 439, 446, 449, 476, 482, 483, 511, 521, 527, 562

Offset: 1

Felix Huber, Mar 26 2025

nonn

A381336(a(n)) is the short leg, a(n) + A381336(a(n)) is the long leg and A381337(a(n)) is the hypotenuse.

			12 is in the sequence because A381336(12)^2 + (A381336(12) + 12)^2 = 36^2 + 48^2 = 60^2 = A381337(12)^2.

Cf. A381336, A381337.

isA382297:=proc(n)
    local k,c,s;
    for k do
        for c from k+n to 2*k+n-1 do
            s:=(n+2*k+c)/2;
            if issqr(s*(s-k)*(s-k-n)*(s-c)) then
                return [k,n+k,c];
            fi
        od
    od;
end proc;
A382297:=proc(n)
    option remember;
    local a;
    if n=1 then
        1
    else
        for a from procname(n-1)+1 do
            if isA382297(a)[1]^2+isA382297(a)[2]^2=isA382297(a)[3]^2 then
                return a
            fi
        od
    fi;
end proc;
seq(A382297(n),n=1..59);

A381336(a(n))^2 + (A381336(a(n)) + a(n))^2 = A381337(a(n))^2.

A381336 a(n) is the smallest k > 0 for which a nondegenerate integer-sided triangle (k, k + n, c >= k + n) with an integer area exists.

Original entry on oeis.org

3, 6, 9, 12, 12, 18, 5, 7, 4, 24, 14, 36, 15, 10, 36, 14, 7, 8, 6, 21, 8, 3, 12, 5, 10, 15, 12, 20, 46, 35, 9, 28, 20, 14, 25, 16, 15, 12, 22, 21, 19, 16, 12, 6, 20, 5, 4, 10, 11, 20, 21, 30, 96, 24, 13, 9, 18, 7, 25, 63, 21, 18, 22, 9, 35, 9, 25, 21, 36, 17, 13

Offset: 1

Felix Huber, Mar 16 2025

nonn

Longest sides c are in A381337.

			a(5) = 12 because the nondegenerate integer-sided triangle (12, 12 + 5, 25 >= 12 + 5) has an integer area (90), and there is no smaller k > 0 than 12 that satisfies this condition.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Heron's Formula.

Cf. A103974, A103975, A188158, A379830, A381337.

A381336:=proc(n)
    local k,c,s;
    for k do
        for c from k+n to 2*k+n-1 do
            s:=(n+2*k+c)/2;
            if issqr(s*(s-k)*(s-k-n)*(s-c)) then
                return k
            fi
        od
    od;
end proc;
seq(A381336(n),n=1..71);

cp's OEIS Frontend

A382297 Indices of right triangles in A381337.

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